Optimal Rates of Convergence for Deconvolving a Density
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Suppose that the sum of two independent random variables X and Z is observed, where Z denotes measurement error and has a known distribution, and where the unknown density f of X is to be estimated. One application is the estimation of a prior density for a sequence of location parameters. A second application arises in the errors-in-variables problem for nonlinear and generalized linear models, when one attempts to model the distribution of the true but unobservable covariates. This article shows that if Z is normally distributed and f has k bounded derivatives, then the fastest attainable convergence rate of any nonparametric estimator of f is only (log n)k/2. Therefore, deconvolution with normal errors may not be a practical proposition. Other error distributions are also treated. StefanskiCarroll (1987a) estimators achieve the optimal rates. The results given have versions for multiplicative errors, where they imply that even optimal rates are exceptionally slow. 1976 Taylor & Francis Group, LLC.
